A clearing system with impatient passengers: asymptotics and estimation in a bus stop model

Abstract

At some location buses stop to pick up the passengers waiting there and leave immediately, empty or occupied. Passenger arrival times as well as bus arrival times form independent renewal processes. Every passenger is willing to wait for a random amount of time before leaving, and every bus takes away all waiting passengers. For this pickup model, we study the distributions of the number of waiting passengers and of the individual sojourn times. The sojourn times lead to a Markov chain embedded in the superposition of the two underlying renewal arrival processes, for which we study its convergence toward stationarity.

Mathematics Subject Classification

Notes

Acknowledgments

We are very grateful to the reviewer and the associate editor for the careful reading and for important observations and corrections that improved the presentation of the paper. In particular, (12) was suggested by the associate editor and replaces a considerably more cumbersome derivation. Offer Kella was supported in part by Grant 1462/13 from the Israel Science Foundation and the Vigevani Chair in Statistics. Wolfgang Stadje was supported by Grant No. 306/13-2 of the Deutsche Forschungsgemeinschaft.

References

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